= Solution
The <characteristic curves> satisfy
$$
-x^2(dy)^2+(dx)^2=0,\qquad\frac{dy}{dx}=\pm\frac1x.
$$
Integrating suggests the <characteristic coordinates>
$$
\boxed{\xi=y+\ln|x|,\qquad\eta=y-\ln|x|.}
$$
Their <Jacobian determinant> is $2/x$, so this is an invertible local <change of variables> whenever $x\ne0$. Write $u(x,y)=U(\xi,\eta)$. The <chain rule> gives
$$
\begin{aligned}
u_x&=\frac{U_\xi-U_\eta}{x},\\
x^2u_{xx}&=U_{\xi\xi}-2U_{\xi\eta}+U_{\eta\eta}-U_\xi+U_\eta,\\
u_{yy}&=U_{\xi\xi}+2U_{\xi\eta}+U_{\eta\eta}.
\end{aligned}
$$
Substitution cancels both unmixed second derivatives and both first derivatives:
$$
-x^2u_{xx}+u_{yy}-xu_x=4U_{\xi\eta}.
$$
The <canonical form of a second-order partial differential equation> is therefore
$$
\boxed{U_{\xi\eta}=1.}
$$
The cancellation of the first derivatives is the key feature of these <logarithmic characteristic coordinates>.
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